Meaning of the sign of the geodesic curvature

In summary, the geodesic curvature sign for a coordinate patch on a surface is determined by the projection of the curve's curvature vector onto the tangent plane and can be either positive or negative depending on the orientation of the surface.
  • #1
Leo Mar
3
0
My question is : what is the meaning of the geodesic curvature sign for a coordinate patch? for a surface? Thank you.
 
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  • #2
Leo Mar said:
My question is : what is the meaning of the geodesic curvature sign for a coordinate patch? for a surface? Thank you.

how do you define geodesic curvature?
 
  • #3
Hello Lavinia, the geodesic curvature of a unit speed curve ϒ is the component of ϒ'' in the direction of S=n×T.
 
  • #4
Leo Mar said:
Hello Lavinia, the geodesic curvature of a unit speed curve ϒ is the component of ϒ'' in the direction of S=n×T.

OK.

Since the surface is oriented the unit tangent to the curve determines an oriented orthogonal direction by 90 degree rotation. This direction may be opposite to the projection of the curve's curvature vector onto the tangent plane. If the projection points opposite to the oriented normal then the geodesic curvature is negative.
 
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What is the meaning of the sign of the geodesic curvature?

The sign of the geodesic curvature represents the direction of the curvature of a curve on a surface at a given point. It is positive if the curve is curving in the same direction as the surface, and negative if it is curving in the opposite direction.

How is the sign of the geodesic curvature calculated?

The sign of the geodesic curvature is calculated using the Frenet-Serret formulas, which take into account the curvature and torsion of the curve as well as the shape of the surface at the given point.

What does a positive sign of the geodesic curvature indicate?

A positive sign of the geodesic curvature indicates that the curve is convex at the given point, meaning that it is curving outward from the surface.

What does a negative sign of the geodesic curvature indicate?

A negative sign of the geodesic curvature indicates that the curve is concave at the given point, meaning that it is curving inward toward the surface.

How is the sign of the geodesic curvature used in applications?

The sign of the geodesic curvature is used in various applications in mathematics, physics, and engineering, particularly in the study of curved surfaces and differential geometry. It is also used in navigation, such as in determining the shortest path between two points on a curved surface.

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